Quantitative C¹-stability of spheres in rank one symmetric spaces of non-compact type

2023/04/05 by Lauro Silini, Silini, Lauro
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2304.02412

openalex publication_date 2023/04/05 · openalex created_date 2023/04/07 · openalex updated_date 2026/07/28

Abstract

We prove that in any rank one symmetric space of non-compact type M∈\ℝ Hⁿ,ℂ H^m,ℍ H^m,\mathbbO H²\, geodesic spheres are uniformly quantitatively stable with respect to small C¹-volume preserving perturbations. We quantify the gain of perimeter in terms of the W^1,2-norm of the perturbation, taking advantage of the explicit spectral gap of the Laplacian on geodesic spheres in M. As a consequence, we give a quantitative proof that for small volumes, geodesic spheres are isoperimetric regions among all sets of finite perimeter.

Related

• A Spectral Characterization Of Geodesic Balls In Non-Compact Rank One Symmetric Spaces
• On Stability and Isoperimetry of Constant Mean Curvature Spheres of \mathbb Hⁿ×\mathbb R and \mathbb Sⁿ×\mathbb R.
• On the harmonic characterization of the spheres: a sharp stability inequality and some of its consequences
• Rigidity of Rank-One Factors of Compact Symmetric Spaces
• Rigidity of compact rank one symmetric spaces